Activity 11.8.5

We will look algebraically calculating and explaining the vector properties of the gradient using the function $f(x,y)=xy-2y^2+\frac{x^2}{2}+2y-x$ at the input $(1,1)$.

1. Calculate the gradient of $f$

2. Calculate $\nabla f (1,1)$.

3. Calculate $D_{\vec{u}} f (1,1)$ for each of the following vectors:

   1. $\vec{u}=\langle -1,0\rangle$
   2. $\vec{u}=\langle \frac{3}{5},-\frac{4}{5} \rangle$
   3. $\vec{u}=\langle -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \rangle$

4. What direction will correspond to the greatest rate of increase for $f$ at the input $(1,1)$? Find the value of the directional derivative in this direction.